Pour all but 1 tablespoon drippings from skillet. Add oil. Reduce heat to medium-high. Add mushrooms and shallots and sauté until mushrooms are tender, about 8 minutes. Season with salt and pepper. Add wine and stir until juices thicken, scraping up browned bits, about 1 minute.

Place 1 duck breast on each of 4 plates. Pour mushroom sauce over, sprinkle with parsley, and serve.